How to cut a rounded hole in a sphere?
This might take considerable more work for more complex shapes but one way would be to use the Shrinkwrap modifier to snap the outline of the shape to your mesh. 



Next, join both shapes and delete the parts of the sphere that your outline contains, now fill in back the rest of the shape.

 

As I said, this method is not perfect and would require some careful editing to make sure the mesh keeps its curved appearance but for simple outlines/shapes it should work fine.

General tips when doing this are to keep your outline clean and preferably as lines as the Shrinkwrap modifier tends to distort faces depending on the surface it is snapping to.



Another way would be to use the Boolean modifier and cut out the shape.

Add the modifier to the object you want to cut and select the object (an extruded Blender logo in this case) and set the operation to Difference with the logo as the target, delete the outside piece of the logo and you should have a nice cut out shape on the sphere.

Once Smooth Shaded this will be very blocky but running a Tris to Quads operation with AltJ and applying an Edge Split modifier should make it look better.



If you need bevelled edges, the boolean modifier leaves very sharp cuts so you should be able to select the jumbled ones and merge and dissolve vertices as needed until you get a clean edge ring without affecting the curvature of your shape much.